Limits of Functions: Interactive Quizzes and Practice Questions

Question 1

Which of the following conditions must be met for a function f(x) to have a limit as x approaches a?

Accepted Answer

The limit from the right and the limit from the left are equal.

Answer

The function is continuous at x = a.

Answer

The function is defined at x = a.

Answer

The function is differentiable at x = a.

Question 2

Which of the following is the most precise definition of a function's limit?

Accepted Answer

The value that the function gets arbitrarily close to as the input variable approaches a specified value

Answer

The value that the function takes on at a specific input

Question 3

Select the limit law used to simplify the limit of a quotient of two functions.

Accepted Answer

The quotient law: lim[f(x)/g(x)] = lim[f(x)] / lim[g(x)]

Answer

The power law: lim[f(x)^(n)] = lim[f(x)]^n

Answer

The product law: lim[f(x)g(x)] = lim[f(x)]lim[g(x)]

Question 4

Find the limit of the function h(x) = (e^x - 1) / x as x approaches infinity.

Accepted Answer

1

Answer

Does not exist

Answer

0

Answer

Infinity

Question 5

Which of the following is a necessary condition for the existence of a finite limit?

Accepted Answer

The left-hand limit and right-hand limit are equal.

Answer

The limit is continuous at the point.

Answer

The limit is finite.

Question 6

Consider the function f(x) = |x|. Find the limit of f(x) as x approaches -2.

Accepted Answer

2

Answer

0

Answer

-2

Answer

Does not exist

Question 7

Which of the following functions has an infinite limit as x approaches 0?

Accepted Answer

f(x) = 1/x

Answer

f(x) = x^2

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Question 8

Determine the vertical asymptotes of the function h(x) = (x² - 9) / (x - 3).

Accepted Answer

x = 3

Answer

The function has no vertical asymptotes.

Answer

x = -3

Answer

y = 3

Question 9

Evaluate the limit of the function f(x) = (x - sin(x)) / (x^2 - π^2) as x approaches π.

Accepted Answer

1/π

Answer

π

Answer

Does not exist

Answer

0

Question 10

Find the horizontal asymptote of the function f(x) = (2x^3 - 5x) / (x^3 + 2).

Accepted Answer

y = 2

Answer

y = 5/2

Answer

The function has no horizontal asymptotes.

Answer

y = -2

Question 11

Determine the vertical asymptote(s) of the function f(x) = (x^2 - 1) / (x - 1).

Accepted Answer

x = 1

Answer

x = 0

Answer

The function has no vertical asymptotes.

Answer

x = -1

Question 12

Find the limit of the function f(x) = e^(x^2) as x approaches ∞.

Accepted Answer

∞

Answer

0

Answer

1

Answer

-∞

Question 13

Find the limit of the function f(x) = (x^2 - 4)^1/3 as x approaches 2.

Accepted Answer

2

Answer

4

Answer

0

Answer

Does not exist

Question 14

If lim_(x->a) f(x) = L and lim_(x->a) g(x) = M, what is lim_(x->a) [f(x) + g(x)]?

Accepted Answer

L + M

Answer

L / M

Answer

L - M

Answer

L * M

Question 15

Which of the following is the correct definition of the limit of a function f(x) as x approaches a?

Accepted Answer

For every ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε.

Answer

For every δ > 0, there exists an ε > 0 such that if 0 < |x - a| < ε, then |f(x) - L| < δ.

Answer

For every ε > 0, there exists a δ > 0 such that if 0 < |f(x) - L| < δ, then |x - a| < ε.

Question 16

What is the limit of the function f(x) = 1/x as x approaches 0?

Accepted Answer

Does not exist

Answer

0

Answer

1

Question 17

What is the limit of the function f(x) = x^2 as x approaches infinity?

Accepted Answer

Infinity

Answer

1

Answer

0

Answer

Does not exist

Question 18

Find the limit of the function f(x) = (x^2 - 4)/(x - 2) as x approaches 2 using L'Hopital's Rule.

Accepted Answer

4

Answer

0

Answer

Does not exist

Answer

2

Question 19

What is the limit of the function f(x) = (e^x - 1)/x as x approaches 0?

Accepted Answer

1

Answer

-1

Answer

0

Question 20

Which of the following statements is true about the limit of a function f(x) as x approaches a?

Accepted Answer

The limit exists if and only if the left-hand limit and the right-hand limit exist and are equal.

Answer

The limit always exists, regardless of the left-hand and right-hand limits.

Answer

The limit exists only if the left-hand limit is equal to the right-hand limit.

Question 21

What is the limit of the function f(x) = (x^3 - 8)/(x - 2) as x approaches 2?

Accepted Answer

12

Answer

4

Answer

8

Answer

Does not exist

Question 22

Which of the following is the definition of the limit of a function at a point 'c'?

Accepted Answer

The limit of a function f(x) as x approaches c is the value L if, for any number ε > 0, there exists a number δ > 0 such that if |x - c| < δ, then |f(x) - L| < ε.

Answer

The limit of a function f(x) as x approaches c is the value L if, for any number δ > 0, there exists a number ε > 0 such that if |f(x) - L| < ε, then |x - c| < δ.

Question 23

Which of the following limit laws expresses the limit of a sum of functions as the sum of their limits?

Accepted Answer

lim (f(x) + g(x)) = lim f(x) + lim g(x)

Answer

lim (f(x) * g(x)) = lim f(x) * lim g(x)

Answer

lim (f(x) / g(x)) = lim f(x) / lim g(x)

Answer

lim (f(x) - g(x)) = lim f(x) - lim g(x)

Question 24

Find the limit of the function f(x) = sin(x) / x as x approaches 0.

Accepted Answer

1

Answer

Undefined

Answer

0

Answer

Infinity

Question 25

Which of the following statements about the existence of a limit at a point is true?

Accepted Answer

If the left-hand limit and the right-hand limit at a point are equal, then the limit exists.

Answer

If the left-hand limit at a point exists, then the limit exists.

Answer

If the left-hand limit and the right-hand limit at a point are different, then the limit exists.

Question 26

Find the horizontal asymptote of the function f(x) = (x^3 - 2x^2 + 3) / (x^2 - 1).

Accepted Answer

y = x

Answer

y = 3

Answer

y = 2

Answer

y = 0

Question 27

Determine the limit of h(x) = (ln(x) - 1) / (x - e) as x approaches e.

Accepted Answer

1/e

Answer

Undefined

Answer

e

Answer

0

Question 28

Find the limit of the function f(x) = (e^(2x) - 1) / (e^x - 1) as x approaches infinity.

Accepted Answer

2

Answer

Infinity

Answer

0

Answer

1

Question 29

Determine the limit of the function g(x) = (sin(5x) - sin(3x)) / (x - 0) as x approaches 0.

Accepted Answer

2

Answer

1

Answer

5

Answer

0

Question 30

Which of the following limit laws is used to evaluate the limit of a sum of functions?

Accepted Answer

Sum Law

Answer

Constant Multiple Law

Answer

Product Law

Answer

Quotient Law

Question 31

What is the limit as x approaches infinity of the function f(x) = (3x^2 + 2x)/(x^2 - 1)?

Accepted Answer

3

Answer

Infinity

Answer

-1

Answer

0

Question 32

Given that lim x->a f(x) = 3 and lim x->a g(x) = -2, what is the limit as x approaches a of the function f(x) * g(x)?

Accepted Answer

-6

Answer

-1

Answer

1

Answer

5

Question 33

What is the limit as x approaches 0 from the right of the function f(x) = 1/x?

Accepted Answer

Infinity

Answer

Does not exist

Answer

Negative Infinity

Answer

0

Question 34

Which of the following statements is true about the limit of a function at a point where it is discontinuous?

Accepted Answer

The limit may exist even if the function is discontinuous at that point.

Answer

The limit always exists if the function is discontinuous at that point.

Answer

The limit never exists if the function is discontinuous at that point.

Question 35

The limit as x approaches 0 of the function f(x) = sin(x)/x is...

Accepted Answer

1

Answer

0

Answer

Infinity

Question 36

What is the limit as x approaches 2 of the function f(x) = (x^3 - 8)/(x - 2)?

Accepted Answer

12

Answer

2

Answer

Does not exist

Answer

8

Question 37

Which of the following functions has a limit as x approaches infinity that is equal to 0?

Accepted Answer

f(x) = 1/x^2

Answer

f(x) = x^2

Answer

f(x) = e^x

Answer

f(x) = sin(x)

Question 38

Evaluate the limit:
lim_(x->0) (sin(x) - x) / x^3

Accepted Answer

1/6

Answer

∞

Answer

1/2

Answer

0

Question 39

Evaluate the limit:
lim_(x->e) (x^(ln(x)) - e)

Accepted Answer

0

Answer

e

Answer

1

Answer

∞

Question 40

Evaluate the limit:
lim_(x->0) (e^(2x) - 1) / x

Accepted Answer

2

Answer

∞

Answer

1

Answer

e^2

Question 41

Use the squeeze theorem to find the limit of the function: f(x) = x sin(1/x) as x approaches 0

Accepted Answer

0

Answer

-1

Answer

1

Answer

π

Question 42

Determine the vertical asymptote(s) of the function: f(x) = (x^2 - 9) / (x - 3)

Accepted Answer

x = 3

Answer

The function has no vertical asymptotes

Answer

x = -3

Answer

y = 3

Question 43

Find the limit of the function: f(x) = |x - 2| as x approaches 2 from the left

Accepted Answer

0

Answer

1

Answer

2

Answer

-2

Question 44

Which of the following is the definition of the limit of a function?

Accepted Answer

For any ε > 0, there exists a δ > 0 such that if 0 < |x - a| < δ, then |f(x) - L| < ε

Answer

The value of the function at x = a

Answer

The value that the function approaches as x gets close to a

Question 45

Determine the limit: lim (x → ∞) (1 + 1/x)^x

Accepted Answer

e

Answer

1

Answer

∞

Answer

0

Question 46

Find the limit of the function f(x) = (x^2 - 4) / (x - 2) as x approaches 2.

Accepted Answer

4

Answer

0

Answer

2

Answer

8

Question 47

What is the limit of the function f(x) = sin(x) / x as x approaches 0?

Accepted Answer

1

Answer

0

Answer

cos(0)

Question 48

Determine the limit of the function f(x) = (e^x - 1) / x as x approaches infinity.

Accepted Answer

1

Answer

0

Answer

infinity

Answer

e

Question 49

Find the limit of the function f(x) = sqrt(x^2 + 1) - x as x approaches infinity.

Accepted Answer

1/2

Answer

infinity

Answer

1

Answer

0

Question 50

Which of the following is the limit of the function f(x) = (x^3 - 8) / (x - 2) as x approaches 2?

Accepted Answer

12

Answer

4

Answer

0

Answer

6

Question 51

Given the function f(x) = 1 / (x^2 - 4), find the limit as x approaches -2.

Accepted Answer

1/4

Answer

infinity

Answer

0

Answer

-1/4

Question 52

What is the limit of the function f(x) = |x - 2| as x approaches 2?

Accepted Answer

0

Answer

1

Answer

2

Answer

4

Question 53

Find the limit of the function f(x) = (x^2 - 9) / (x + 3) as x approaches -3.

Accepted Answer

0

Answer

6

Answer

infinity

Answer

-6

Question 54

Determine the limit of the function f(x) = (1 - cos(x)) / x^2 as x approaches 0.

Accepted Answer

1/2

Answer

0

Answer

infinity

Answer

1

Question 55

Find the limit of the function f(x) = (ln(1 + x) - ln(2)) / x as x approaches 0.

Accepted Answer

1/2

Answer

1

Answer

infinity

Answer

0